Discrete Mathematics 2010-2022

Random phenomena often appear in dynamical systems that we aim to analyse and to control.
Mathematics serves to describe these random dynamical systems by stochastic differential equations (SDEs).
In many cases the coefficients of these SDEs lack regularity properties that are assumed in the classical literature on numerical methods for SDEs.
For example, when solving stochastic control problems by simulation one has to take into account that the control might depend on the controlled process in a discontinuous manner.

Motivated by this problem we study existence, uniqueness, and strong convergence rates of numerical methods for certain SDEs with non-globally Lipschitz coefficients.

For a sparse random matrix model with given numbers of non-zero entries in the rows and columns we identify a sufficient and very nearly necessary condition for the matrix having full row rank whp. The condition comes exclusively in terms of the degree distributions and holds for matrices with independent entries over finite fields. As an implication we obtain a full rank condition for random $0/1$ matrices over the rationals. Joint work with Max Hahn-Klimroth, Joon Lee, Noela Muller, Maurice Rolvien.

We extend a recent argument of Kahn, Narayanan and Park about the threshold for the appearance of the square of a Hamilton cycle to other spanning structures. In particular, for any spanning graph, we give a sufficient condition under which we may determine its threshold. As an application, we find the threshold for a set of cyclically ordered copies of $C_4$ that span the entire vertex set, so that any two consecutive copies overlap in exactly one edge and all overlapping edges are disjoint. This answers a question of Frieze. We also determine the threshold for edge-overlapping spanning $K_r$-cycles.